


Expectation and Maximization

by eve11



Category: Doctor Who, Doctor Who (2005)
Genre: Alien Planet, Gen, Mathematics, One Shot
Language: English
Status: Completed
Published: 2011-06-12
Updated: 2011-06-12
Packaged: 2017-10-20 08:57:21
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 2,972
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/211004
Author URL: https://archiveofourown.org/users/eve11/pseuds/eve11
Summary: <blockquote class="userstuff">
              <p>"Rose Tyler, you've faced clockwork droids, Slitheen and Cybermen, not to mention five different types of zombies and a megalomaniacal nightmare better known as the Emperor of the Daleks, and you're telling me you're afraid of some maths?"</p>
            </blockquote>





	Expectation and Maximization

"Alcarasthandimal?" Rose's hand hesitated at the TARDIS door and she turned around, questioning. The Doctor looked at her appraisingly as he sauntered down the ramp from the center console.

"Close enough, not bad pronunciation given the lack of a forked tongue."

She smiled, testing her teeth with the tip of her non-forked tongue as he caught up to her. He donned his trench coat, raised his eyebrows and indicated the door.

"And the sky is silver?" she asked.

"Of a sort. Lots of nebulae and dust, with little atmospheric interference."

She narrowed her eyes. "But it's safe, yeah?"

"Perfectly." He rocked on his heels. "Go on. See for yourself."

In the space of a second she abandoned her caution and cast open the door. She was ten paces ahead on the level ground, spinning and staring up at the sky, when he stepped out and closed the door behind him.

"Oh, it's beautiful. I love it!" Rose cried, arms aloft. She wrested her gaze from the glittering sky, back over to the blue box. "Doctor--" she started, but stopped. Her hands fell.

The Doctor was stern-faced, hands in his pockets, concentrating. He hadn't moved more than a foot beyond the door. After a moment his eyes focused on her, and he blinked.

"Sorry," he said. "Sorry, we can't stay."

"What? But you said the atmosphere was safe."

"Oh yes, perfectly. It's not that." He took a step and swayed, holding a hand out to the TARDIS for support.

Rose shot back to his side, the silver sky abandoned. "Are you--?"

"Fine, fine," he said, waving her off. He steadied himself and studied his surroundings with a furrowed brow. "It's the gradient here; it's . . . too flat." Dissatisfied, he frowned and tried another word. "The gradient is weak. A bit disorienting at first, but perfectly natural. Mind you, it's also a big blinking neon sign that says, 'Time travellers, stay away'. We have to go, leave this place to the natural order."

"Why?" Rose asked.

The Doctor fished the TARDIS key from his pocket. "Haven't a clue what's causing it, but this place, this time-- well, it's not unstable so much as unconstrained. Wilderness. Low likelihood actions with highly variable consequences, wandering off to who knows where and when. Blimey, can't you feel--?" He stopped mid-sentence, his consternation melting into a sad-eyed smile as he studied her face. "But of course you can't."

Rose scuffed a shoe on the cherry-red grass, and each of them was reminded again that in addition to everything else, the other was still irrevocably alien. The moment passed, and Rose took a breath and met the Doctor's gaze.

"Gradient, you said. What's that, some kind of forcefield or something?"

"Sort of. It's a directional vector of change. In this case, relative to the likelihood of events in our frame of reference. Well," he added, "sometimes it's the negative log-likelihood, if you want to be precise. Well--"

"Oh no, none of that," Rose interrupted. "Can't say I followed it, but it sounds suspiciously like algebra."

"Probability measure," the Doctor corrected. "And algebra, I suppose. And calculus."

"Ugh, I don't want to go near it." Her teasing smile almost reached her eyes. The Doctor stood straight, taken aback.

"Rose Tyler, you've faced clockwork droids, Slitheen and Cybermen, not to mention five different types of zombies and a megalomaniacal nightmare better known as the Emperor of the Daleks, and you're telling me you're afraid of some maths?"

"No, but it's . . . I can't understand it. Not all that."

"Nonsense. It's all just a bit of hill-climbing."

"Get out." She chucked his shoulder. He held up a hand.

"No, no. More properly, it's quite the reverse." His eyes danced as he opened the TARDIS door. "Come on. I'll show you."

\---------

"Here we are, then," the Doctor said, following Rose out the door. "Watch your step."

The TARDIS had set down on a vast plain under a cobalt sky. Olive green not-grass made a springy carpet of strong, intertwining fibers under their feet; Rose couldn't help but make a few experimental hops before shielding her eyes and looking toward the horizon.

"So, where is here?" she asked.

They were on high ground, at the lip of a bowl-shaped depression half the size of a football pitch. And as far as the view permitted, the plain was pocked with them--semi-smoothed ridges and irregular craters like steep hills in relief.

"God's Knuckles," the Doctor answered.

She gave him a look, like he was pulling her leg. He looked back with mock seriousness.

"That's the closest translation, cross my heart," he said. "The local population's creation myth said the world was shaped and risen from a ball of dough, only in some places it didn't rise as it should have, before it set."

"So that's the marks God made in the world with his knuckles--"

"While kneading the dough, exactly. Very clever and creative of them, don't you think? 'Course, the Carvarians don't have knuckles per se." He brought a thumb to his lips, contemplating the sky for a moment. "More like nodules, maybe. But 'God's Nodules' really doesn't convey the right image at all now, does it?" He trailed off when he noticed Rose suppressing a smile. "Where was I?"

"Maths?" she suggested.

The Doctor cleared his throat. "Ah yes! Quite right. The point is, that we're here because it's a perfect place to explain modes."

"Modes?"

"Modes. Stable event structures in time and space, relative to the observer's experience."

Rose frowned. "What do you mean?"

"Imagine every point of latitude and longitude on this plain is an event in time and space." He held an arm to her waist, directing her gaze to a crater several hundred meters distant. "See there? There, World War Two never happens on Earth. And there," another depression, farther away, "You and I never meet because neither of us is ever born. And way over there," a crater kissing the horizon, "Humans evolve with an extra set of arms, giant space-faring snakes block out the sun, and your Mum greets me with a smile, a four-armed hug and an invite to tea whenever we stop by."

"Oh, she doesn't hate you that much."

"Mmm," he said absently. "Only on days ending with 'y'. Luckily, in that mode, Wednesday, Thursday and Saturday all have a silent 'e' tacked at the end."

Rose's laugh echoed across the plain, and she turned to face him. "So they're all different universes?"

The Doctor shook his head. "Same universe, different configurations. The ridges and valleys are a measure of likelihood over the events in time and space. History reorganizes itself into another stable structure at each mode."

"All right. I think I understand." She looked down into the bowl in front of them and smiled wickedly. "So? What's this one, then?"

He smiled back, excitement gleaming in his eye. "Don't you recognize it? Powell Estate, beans on toast, shop window dummies? Can't you feel it pulling us in?" He took her hand and propelled her forward. She let out a surprised cry, and then they bounded down over the lip into the hollow, the resilient ground fueling their steps and swallowing the echo of his words.

"This one's our very own local maximum!"

\--------------

"This is a maximum?" Rose asked.

They were at the very bottom of the crater. The Doctor surveyed the green walls surrounding them, and Rose stood very still, her eyes closed in concentration.

"Minimum altitude, maximum likelihood," the Doctor explained. "What do you feel, here? What does the curve of the land tell you?"

"It's--I don't know. Feels heavy, like an anchor, with the walls all around."

The Doctor took hold of her shoulder and gently moved her. She swayed, righting herself easily.

"Very good," he said. "You're stable. Think of this place, the bottom of this bowl, as where you belong and what you know. Now, me"--he took her hand again, and she opened her eyes--"Ah, keep your eyes closed. I come along in the TARDIS and pluck you out of space and time. Off we go!"

He picked her up and spun her around, and as soon as her feet touched ground again they set off in a meandering, dizzying path across the crater. Rose giggled and shrieked, but kept her eyes closed as he brought them both to a stop half-way up the steep side of the bowl. She veered a bit, shot out a hand and held fast to his lapel.

"We've landed," he said.

"Can I open my eyes yet?" she asked as she stabilized herself.

"Not yet, that's cheating. First, tell me, which way to the mode? What's the easiest way to get back to what we know?"

Rose furrowed her brow, then oriented herself and unerringly extended a hand. "That way," she said.

"Fantastic! Open your eyes."

She opened her eyes and found herself pointing directly down to the bottom of the bowl. "Oh my god," she said, "I'm like a homing pigeon."

"That's the gradient at this point," the Doctor said. "The direction and strength of change. What you feel here is very strong. Likelihood is gravity, pulling you toward the mode, steepest along the gradient. And the wall at your back is telling you what events take you farther away. What isn't supposed to happen. Go too far in that direction, and you're bound for trouble."

"That's how you know?" Rose asked, amazed. "How we don't get turned around? You can feel this, every time we land?"

"Every time. It's hard to get lost on the side of a bowl, isn't it?" He took a step along the steepest downward path, Rose following behind. "The gradient will always get us back to our stable mode. And toward the bottom where the gradient is weaker, we have the reflection of the walls to guide us."

"But Doctor," Rose said, surveying her position with a critical eye. "There's another way we can go, isn't there? Without getting closer or farther away."

He paused, and looked back at her with glee. "How do you mean?" he encouraged.

"I know the gradient goes this way." She pointed down toward the mode, and then turned to her right and took a step, tracing out an arc along the side of the bowl. "But there's no steepness at all in this direction. No change; it's completely flat."

"Level curves!" the Doctor exclaimed. "Brilliant."

"And what are they?"

"The best direction in which to meddle," he said, leading her along the level curve. "Change of events, without changing the likelihood. And it's easy to find this direction too, if you know the gradient."

"It is?"

"'Course it is. You did it, didn't you?"

"I did? I mean, I went--" she stopped, snapping her fingers and trying to think of the word. "Perpendicular! Ninety degrees to the direction of the gradient. Yeah?"

The Doctor turned around. "Rose Tyler, you've no business being afraid of maths. You're a natural. Now, do you fancy something a bit more dangerous?"

\----------------

"All right. What do you feel?"

"Confused."

"As well you should be. This is a very low-likelihood, unstable event. Like, oh, crossing your own time stream or something."

They were perched atop a high, meandering wrinkle, branching out along the plain. There were perhaps three feet of gentle slope along the wrinkle before the walls of craters and bowls loomed forth, forming ledges and steep drops. Their crater--the one they had just left--rushed away behind them, the empty dome leaving an impression of negative space at their backs, the familiar mode a small point well below the level of their feet.

"I can't feel any walls," Rose said, clutching the Doctor's arm tightly.

"But you can feel the modes, right? Nearby?"

"Yeah, some."

"Which way does the gradient say to go?" he asked.

Eyes closed, Rose felt the slope of the hill underneath her. "It's harder to tell. But I think it says that way." She pointed along the direction of steepest descent and opened her eyes. She was dismayed to see it leading to an entirely different bowl than the one they'd left.

The Doctor looked down at the new crater. "Hm. Pigs with wings. That just won't do. Don't think you'd like that mode at all."

She looked down. "But once we go off in that direction, the gradient won't ever lead us back to ours, will it?"

"No," he said. "At an unstable point, you run the risk of history as you know it reordering itself. And what do you suppose would happen if this ridge were sharply peaked, like a rooftop?"

"Well, you couldn't even stand on the top," Rose said.

"And the gradient at that point?"

Rose thought about this for a few moments, narrowing her eyes. "I don't know. Seems like it would depend on what side of the peak you were on."

"Exactly," the Doctor said. "If you're right at the top and it's peaked enough, the gradient isn't well defined. And even if it's not peaked, you still can't orient with it, as you've discovered. My people"--he took a breath and sighed, surveying the landscape with dark eyes--"well, they didn't approve of coming up here, that's for sure."

Rose turned to him. "But you've been here--places like here, before? What can you do, if you can't follow the gradient once you're up here?"

He put his hands in his pockets. "Oh, this and that. Mostly, you've got to remember the direction you came from. It takes practice, good balance and very small steps."

"And this is what you felt at Alcarasthandimal? This is why we couldn't stay?"

"Nah," he said, rocking on his heels in defiance of the drop on either side. "Ridges I can deal with. I'm quite the expert at navigating them. And," he tugged her shoulder and she yelped; they stumbled off the wrinkle back down onto the slope of their bowl, "at falling in the right direction. No, Alcarasthandimal--and good for you for remembering the name, by the way--Alcarasthandimal was probabilistic no-man's land."

"How?" Rose asked. "What did it feel like?"

The Doctor gave her a hand up, shielding his eyes and scanning the horizon until he focused on a point somewhere in the distance.

"Come on," he said. "We'll need the TARDIS for this one."

\-------------

The TARDIS doors opened, and the two companions stepped tentatively out. Rose had her eyes closed, following the Doctor's lead with her hands at his elbow.

"Small steps," the Doctor said, and they moved a few quiet paces from the comfort of the TARDIS door. "Now. What can you feel?"

Rose concentrated, but there was nothing. No reflection of walls, no slope under her feet, not even the whistling absence of a retreating mode as there had been on the ridge. It was just . . . flat. Completely, totally, utterly flat.

"I--" she started, and gasped as a wave of dizziness swept over her. Instinctively, she opened her eyes.

The view hit her like a locomotive. Glistening, sliver sky overhead. Soft red grass underneath her on a perfectly level plain. God's Knuckles was gone; this was Alcarasthandimal. Her knees wobbled.

"Doctor--" she said, suddenly swaying.

"Careful." The Doctor caught her arm with his free hand, helping her right herself. "I'm none too sturdy myself, here."

"Oh my god," Rose said, studying the landscape. Her heart pounded in her chest. "I've no idea how to get back."

The Doctor rubbed at his eyes. "We're on a high plateau. On Alcarasthandimal, yes, but on the likelihood surface too. I know which way we came from, and I can get us back as long as I don't lose that sense of direction, but it's dangerous to wander through events here. Too easy to get turned around, without the gradient."

Understanding blossomed across Rose's face. "We can't stay," she said. "We can't. Who knows what mode we'd come back to if we did?"

The Doctor took her hand. "Wise words, Rose Tyler. Wise indeed."

\----------------

The TARDIS wheezed its way through the vortex as it had a hundred times before, spiriting its two passengers away from Alcarasthandimal and toward more constrained areas of time and space. The Doctor noncommittally flicked a few switches. Rose sat back on the tattered console seat, staring at the column as it hemmed and hawed. After a while, she spoke.

"It's always flat for me, isn't it?" she asked.

"Pardon?"

"Likelihood." She waved a hand. "Events. I--I could get lost on the side of a bowl, because I can't feel a thing."

"You're human." He pressed another button and stepped back, abandoning the console and scanning the familiar, sloping walls of the room. "Humans can't feel the gradient. They don't have the capability."

She looked down at her hands. "It's like being blind."

"Would be for me, if I lost it. I'd have to settle somewhere. Park the TARDIS, fix myself in space." His tone was offhand, but Rose saw the alien fright in his eyes. Something, some feeling, rushed up from the pit of her stomach, dizzying her like the view of that silver sky. She just let it pass.

"Or," the Doctor continued, "I could let go of everything. Every last thing, and keep on going." He looked down at her. "Good thing I'm not planning on losing that bit of my brain, then, isn't it?"

"I don't know," Rose said, a smile gracing her lips. "The way you drive this thing, are you sure it's not going in your old age?"

The Doctor laughed. "You know what I love about humans?" he asked.

"What's that?"

"That I can pluck you up, hurtle you across space and time, and you'll still be able to run out of those doors and enjoy the sky on Alcarasthandimal."


End file.
